ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltexpri GIF version

Theorem ltexpri 7433
Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
Assertion
Ref Expression
ltexpri (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexpri
Dummy variables 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . . . . 8 ((𝑦 = 𝑢𝑧 = 𝑣) → 𝑧 = 𝑣)
21eleq1d 2208 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝑧 ∈ (2nd𝐴) ↔ 𝑣 ∈ (2nd𝐴)))
3 simpl 108 . . . . . . . . 9 ((𝑦 = 𝑢𝑧 = 𝑣) → 𝑦 = 𝑢)
41, 3oveq12d 5792 . . . . . . . 8 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝑧 +Q 𝑦) = (𝑣 +Q 𝑢))
54eleq1d 2208 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 +Q 𝑦) ∈ (1st𝐵) ↔ (𝑣 +Q 𝑢) ∈ (1st𝐵)))
62, 5anbi12d 464 . . . . . 6 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵)) ↔ (𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))))
76cbvexdva 1901 . . . . 5 (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵)) ↔ ∃𝑣(𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))))
87cbvrabv 2685 . . . 4 {𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))} = {𝑢Q ∣ ∃𝑣(𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))}
91eleq1d 2208 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝑧 ∈ (1st𝐴) ↔ 𝑣 ∈ (1st𝐴)))
104eleq1d 2208 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 +Q 𝑦) ∈ (2nd𝐵) ↔ (𝑣 +Q 𝑢) ∈ (2nd𝐵)))
119, 10anbi12d 464 . . . . . 6 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵)) ↔ (𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))))
1211cbvexdva 1901 . . . . 5 (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵)) ↔ ∃𝑣(𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))))
1312cbvrabv 2685 . . . 4 {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))} = {𝑢Q ∣ ∃𝑣(𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))}
148, 13opeq12i 3710 . . 3 ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ = ⟨{𝑢Q ∣ ∃𝑣(𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))}, {𝑢Q ∣ ∃𝑣(𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))}⟩
1514ltexprlempr 7428 . 2 (𝐴<P 𝐵 → ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ ∈ P)
1614ltexprlemfl 7429 . . . 4 (𝐴<P 𝐵 → (1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) ⊆ (1st𝐵))
1714ltexprlemrl 7430 . . . 4 (𝐴<P 𝐵 → (1st𝐵) ⊆ (1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)))
1816, 17eqssd 3114 . . 3 (𝐴<P 𝐵 → (1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (1st𝐵))
1914ltexprlemfu 7431 . . . 4 (𝐴<P 𝐵 → (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) ⊆ (2nd𝐵))
2014ltexprlemru 7432 . . . 4 (𝐴<P 𝐵 → (2nd𝐵) ⊆ (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)))
2119, 20eqssd 3114 . . 3 (𝐴<P 𝐵 → (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (2nd𝐵))
22 ltrelpr 7325 . . . . . . 7 <P ⊆ (P × P)
2322brel 4591 . . . . . 6 (𝐴<P 𝐵 → (𝐴P𝐵P))
2423simpld 111 . . . . 5 (𝐴<P 𝐵𝐴P)
25 addclpr 7357 . . . . 5 ((𝐴P ∧ ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ ∈ P) → (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) ∈ P)
2624, 15, 25syl2anc 408 . . . 4 (𝐴<P 𝐵 → (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) ∈ P)
2723simprd 113 . . . 4 (𝐴<P 𝐵𝐵P)
28 preqlu 7292 . . . 4 (((𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) ∈ P𝐵P) → ((𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵 ↔ ((1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (1st𝐵) ∧ (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (2nd𝐵))))
2926, 27, 28syl2anc 408 . . 3 (𝐴<P 𝐵 → ((𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵 ↔ ((1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (1st𝐵) ∧ (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (2nd𝐵))))
3018, 21, 29mpbir2and 928 . 2 (𝐴<P 𝐵 → (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵)
31 oveq2 5782 . . . 4 (𝑥 = ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ → (𝐴 +P 𝑥) = (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩))
3231eqeq1d 2148 . . 3 (𝑥 = ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ → ((𝐴 +P 𝑥) = 𝐵 ↔ (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵))
3332rspcev 2789 . 2 ((⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ ∈ P ∧ (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵) → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
3415, 30, 33syl2anc 408 1 (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  wrex 2417  {crab 2420  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7100   +Q cplq 7102  Pcnp 7111   +P cpp 7113  <P cltp 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-iplp 7288  df-iltp 7290
This theorem is referenced by:  lteupri  7437  ltaprlem  7438  ltaprg  7439  ltmprr  7462  recexgt0sr  7593  mulgt0sr  7598  map2psrprg  7625
  Copyright terms: Public domain W3C validator